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The scene with major Fambrough is very entertaining, but leaves some out from the script I found. In it, Fambrough acts crazy throughout, but gets decidedly worse right after Dunbar leaves, right before the suicide shot that Dunbar hears as the wagon rolls out of town.

In the script, Fambrough leaves the office naked, except for a plumed hat, his sword, and his revolver. (As written, this reminds me of the incident with Edgar Allen Poe, who arrived to duty similarly once, getting himself expelled from West Point.) This is just after the drink he takes when Dunbar leaves which Dunbar sees through the window. He is pursued by Lieutenant Elgin, who wants to return him to the office. Elgin hears an aside comment from the sargeant who calls Fambrough "crazy", just before Fambrough raises the revolver to protect himself from Elgin. Elgin continues to carefully approach, but doesn't get there in time for the suicide which occurs on the parade ground.

Earlier script notes describe Fambrough as having "sad, swollen eyes. He is an army lifer passed over too many times for promotion and right now does not look like a well man" and, "Sweat has broken out all over him. His grooming is awful. His hands are trembling slightly. Something is very wrong with him."

Earlier, the clinking of the glass in the drawer intimates at alcoholism. There is also emphasis about the childish flourishes during filling the form and the signing, with the giggling.

The way I saw it, Fambrough was a sad little king of a sad little kingdom. He probably envisioned his work to be to bring civilization to the frontier, but the frontier kept moving west and he got left behind. When he encounters Dunbar, he is confronted with unbearable truths. His frontier kingdom is an illusion, his power is meaningless, and before him stands a man on a true path to become a human being. Before him stands a king. Faced with this, Fambrough's life no longer has meaning.

an anonymus editor posted this: Terri is on the right track! The Maj. Farmbrough character may have been based on real U.S. Cavalry officer, Ranald MacKenzie, a much-decorated Civil War officer who later achieved even more fame fighting Indians. MacKenzie was eventually retired early because of mental instability caused by terminal venereal disease.
–
oersOct 26 '12 at 11:49

It's been years since I saw Dances With Wolves, but I remember my "only watches true stories" father-in-law asking me the same question about that scene.

Pretty sure it's an indication that Dunbar has gotten so far away from the center of civilization that the only people crazy enough to go out this far are ... well ... crazy. Or they go crazy from being out that far.

Also, the major is the only person who knows that Dunbar is out their by himself in the frontier. When he kills himself, Dunbar is truly alone. Especially troubling since, if I remember correctly, Dunbar writes in his diary that he expecting back and up supplies to arrive at his one-man fort -- we the audience no there is no one coming.

In the voice over provided by Dunbar/Costner as he arrives at Fort Hayes, Kansas he states, "The bloody slaughter continues in the East..." Though Maj. Fambrough is obviously an unstable alcoholic, I've always thought his reference to "The King is dead; long live the King" may mean that word has arrived of Lincoln's assassination. If correct, then the war has been over several weeks.

The author of the screen play was Michael Blake, who, two years earlier, at urging of Kevin Costner, wrote the novel Dances with Wolves. I know the question is about the movie, but I thought I'd talk about the original novel.

Below is cited from the novel:

Lieutenant Dunbar had it pegged better than he knew, because this
major had, for some time, clung to sanity by the slenderest thread,
and the thread and finally snapped ten minutes before Lieutenant
Dunbar walked into the office. The major had sat calmly at his desk,
hands clasped neatly in front of him, and forgotten his entire life.
It had been a powerless life, fueled by the pitiful handouts that come
to those who serve obediently but make no mark. But all those years of
being passed over, all the years of lonely bachelorhood, all the years
of struggle with the bottle, had vanished as if by magic. The bitter
grind of Major Fambrough's existence had been supplanted by an
imminent and lovely event. He would be crowned King of Fort Hays
sometime before supper.

A few paragraphs later:

"I'm in a generous mood and I will grant your boon. A wagon loaded
with goods of the realm leaves shortly. Find the peasant who calls
himself Timmons and ride with him." .... "My seal will guarantee your
safe conduct through one hundred and fifty miles of heathen
territory."

Finally, several pages later:

Old Major Fambrough, a midlevel administrator with a lackluster
record, had gone off his rocker. He had stood one afternoon in the
middle of the parade ground, jabbering incoherently about his kingdom
and asking over and over for his crown. The poor fellow had been
shipped east just a few days ago.

In the novel the major is described twice as "the liquor-breathed major." That's all. he had a "lonely life."

I believe that it was all of the above combined with absolutely no one to talk to. The enlisted soldiers are all described in ways as to indicate they are more or less illiterate. Being the Camp Commander ensured that his lonely life would remain so.

I found it interesting that this scene takes place so soon after Dunbar's suicide attempt. It felt like he had entered another world, and that if he stayed, this is what he had to look forward to. "There, by the grace of God..."

Well, the reason that it's in the storyline is because there were only two persons who knew where Dunbar was going. The Major (who committed suicide) and the guy with the mules (who gave Dunbar a ride and later got killed by Indians.)
So that would explain how it was possible, that no one would know Dunbar was there - leaving him time to develope his relationship with the Souix.
However, it seems that Major Fambrough obviously had a neurolgial disease. But I think it was brought on by alcoholism.
If you go back and see him in the begining... he's searching for something in his desk draw. When he opens it up... you can hear a bunch of bottles rattling in the drawer.
So I would suspect he had a problem with alcoholism.

Just saw the movie again last night.. My impression is that the Major (Who committed suicide) had known that the post he was sending Dunbar too was ransacked, and the Major had been profiting on the supplies for that post.

Along comes Dunbar, and the Major thought the jig-was-up.. So he offed himself.